Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 627986315 - Original comment: ** |
Wikispaces>TallKite **Imported revision 627986443 - Original comment: deleted much of the mathematical proofs section** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-25 04: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-25 04:33:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>627986443</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>deleted much of the mathematical proofs section</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
| Line 483: | Line 483: | ||
<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span> | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span> | ||
This is a lot of math, but it only needs to be done once for each pergen! | This is a lot of math, but it only needs to be done once for each pergen! | ||
It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation, Large enharmonics occur when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, sixth-4th, ninth-4th, eighth-5th, etc. This is known as the "half-step glitch". Large enharmonics also occur with other pergens. | |||
==Alternate enharmonics== | ==Alternate enharmonics== | ||
| Line 953: | Line 955: | ||
to do: | to do: | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | ||
http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments | http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments | ||
http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments | http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments | ||
=== | ===Notation guide PDF=== | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens. | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens. | ||
| Line 1,005: | Line 1,006: | ||
==Various proofs (unfinished)== | ==Various proofs (unfinished)== | ||
Although not rigorously proven, the two false-double tests have been empirically verified by alt-pergenLister. | |||
The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping. | The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping. | ||
| Line 1,032: | Line 1,035: | ||
C = (u, v, w) | C = (u, v, w) | ||
Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A: | |||
Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. | |||
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | ||
| Line 1,040: | Line 1,041: | ||
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C) | Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C) | ||
Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular **//[I think, not sure | Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular **//[I think, not sure]//**, and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have: | ||
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | ||
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) | 3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) | ||
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C) | Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C) | ||
For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test. | For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test. | ||
Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? | |||
Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b. | Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b. | ||
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q) | a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q) | ||
| Line 1,068: | Line 1,056: | ||
Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb). | Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb). | ||
| Line 1,089: | Line 1,075: | ||
GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b') | GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b') | ||
Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen. | Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen. | ||
To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts | To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts | ||
| Line 1,103: | Line 1,087: | ||
P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G | P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G | ||
==Miscellaneous Notes== | ==Miscellaneous Notes== | ||
| Line 1,175: | Line 1,115: | ||
gedra | gedra | ||
edomapping | edomapping | ||
__**Staff notation**__ | __**Staff notation**__ | ||
| Line 1,240: | Line 1,179: | ||
<!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div> | <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div> | ||
<!-- ws:end:WikiTextTocRule:142 --><!-- ws:start:WikiTextTocRule:143: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*- | <!-- ws:end:WikiTextTocRule:142 --><!-- ws:start:WikiTextTocRule:143: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notation guide PDF">Notation guide PDF</a></div> | ||
<!-- ws:end:WikiTextTocRule:143 --><!-- ws:start:WikiTextTocRule:144: --><div style="margin-left: 3em;"><a href="#toc25"> </a></div> | <!-- ws:end:WikiTextTocRule:143 --><!-- ws:start:WikiTextTocRule:144: --><div style="margin-left: 3em;"><a href="#toc25"> </a></div> | ||
<!-- ws:end:WikiTextTocRule:144 --><!-- ws:start:WikiTextTocRule:145: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-alt-pergenLister">alt-pergenLister</a></div> | <!-- ws:end:WikiTextTocRule:144 --><!-- ws:start:WikiTextTocRule:145: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-alt-pergenLister">alt-pergenLister</a></div> | ||
| Line 3,018: | Line 2,957: | ||
<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | ||
This is a lot of math, but it only needs to be done once for each pergen!<br /> | This is a lot of math, but it only needs to be done once for each pergen!<br /> | ||
<br /> | |||
It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation, Large enharmonics occur when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, sixth-4th, ninth-4th, eighth-5th, etc. This is known as the &quot;half-step glitch&quot;. Large enharmonics also occur with other pergens.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:84 -->Alternate enharmonics</h2> | <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:84 -->Alternate enharmonics</h2> | ||
| Line 6,616: | Line 6,557: | ||
<br /> | <br /> | ||
to do:<br /> | to do:<br /> | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8153:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --><a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><!-- ws:end:WikiTextUrlRule:8153 --><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8154:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --><a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><!-- ws:end:WikiTextUrlRule:8154 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:108:&lt;h3&gt; --><h3 id="toc24"><a name="Further Discussion-Supplemental materials*- | <!-- ws:start:WikiTextHeadingRule:108:&lt;h3&gt; --><h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notation guide PDF"></a><!-- ws:end:WikiTextHeadingRule:108 -->Notation guide PDF</h3> | ||
<br /> | <br /> | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8155:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:8155 --><br /> | ||
<br /> | <br /> | ||
Screenshots of the first 2 pages:<br /> | Screenshots of the first 2 pages:<br /> | ||
| Line 6,634: | Line 6,574: | ||
<br /> | <br /> | ||
Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8156:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:8156 --><br /> | ||
<br /> | <br /> | ||
The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | ||
| Line 6,662: | Line 6,602: | ||
<!-- ws:start:WikiTextHeadingRule:114:&lt;h2&gt; --><h2 id="toc27"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:114 -->Various proofs (unfinished)</h2> | <!-- ws:start:WikiTextHeadingRule:114:&lt;h2&gt; --><h2 id="toc27"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:114 -->Various proofs (unfinished)</h2> | ||
<br /> | <br /> | ||
Although not rigorously proven, the two false-double tests have been empirically verified by alt-pergenLister.<br /> | |||
<br /> | |||
The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br /> | The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br /> | ||
<br /> | <br /> | ||
| Line 6,688: | Line 6,630: | ||
C = (u, v, w)<br /> | C = (u, v, w)<br /> | ||
<br /> | <br /> | ||
Here u, v and w are integers. If GCD (u, v, w) &gt; 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:<br /> | |||
Here u, v and w are integers. If GCD (u, v, w) &gt; 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.<br /> | |||
<br /> | <br /> | ||
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | ||
| Line 6,696: | Line 6,636: | ||
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)<br /> | Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)<br /> | ||
<br /> | <br /> | ||
Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| &gt; 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular <strong><em>[I think, not sure | Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| &gt; 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular <strong><em>[I think, not sure]</em></strong>, and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:<br /> | ||
<br /> | <br /> | ||
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | ||
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)<br /> | 3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)<br /> | ||
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)<br /> | Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)<br /> | ||
<br /> | <br /> | ||
For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r &gt; 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.<br /> | For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r &gt; 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.<br /> | ||
<br /> | <br /> | ||
Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?<br /> | |||
<br /> | <br /> | ||
Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.<br /> | Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.<br /> | ||
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)<br /> | a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)<br /> | ||
| Line 6,724: | Line 6,651: | ||
<br /> | <br /> | ||
Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).<br /> | Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 6,745: | Line 6,670: | ||
GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')<br /> | GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')<br /> | ||
Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.<br /> | Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.<br /> | ||
<br /> | <br /> | ||
To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts<br /> | To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts<br /> | ||
| Line 6,759: | Line 6,682: | ||
P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G<br /> | P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:116:&lt;h2&gt; --><h2 id="toc28"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:116 -->Miscellaneous Notes</h2> | <!-- ws:start:WikiTextHeadingRule:116:&lt;h2&gt; --><h2 id="toc28"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:116 -->Miscellaneous Notes</h2> | ||
| Line 6,831: | Line 6,710: | ||
gedra<br /> | gedra<br /> | ||
edomapping<br /> | edomapping<br /> | ||
<br /> | <br /> | ||
<u><strong>Staff notation</strong></u><br /> | <u><strong>Staff notation</strong></u><br /> | ||